In this thesis, I have explored the different approaches towards proving Artin's `primitive root' conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific set of 13 distinct numbers such that for at least one of them, the conjecture is true. Along the same line, using sieve theory, D. R. Heath-Brown reduced this set down to 3 distinct primes in the year 1986. This is the best unconditional result we have so far. In the first part of this thesis, we will review the sieve theoretic approach taken by Gupta-Murty and Heath-Brown. The second half of the thesis will deal with the elliptic curve analogue of the Artin's conjecture, which is also known as the Lang-Trotter conjecture. Lang and Trotter proposed the elliptic curve analogue in 1977, including the higher rank version, and also proceeded to set up the mathematical formulation to prove the same. The analogue conjecture was proved by Gupta and Murty in the year 1986, assuming the generalized Riemann hypothesis, for curves with complex multiplication. They also proved the higher rank version of the same. We will discuss their proof in details, involving the sieve theoretic approach in the elliptic curve setup. Finally, I will conclude the thesis with a refinement proposed by Gupta and Murty to find out a finite set of points on the curve such that at least one satisfies the conjecture.

Throughout the history of college football, there have been efforts to reform the system and stop improprieties, yet conflict between gaining academic and athletic prowess at colleges remained a central theme. In the 1980s, the Jan Kemp trial involving the University of Georgia demonstrated this clash between revenue-generating athletics and academic integrity. This historical study is an in-depth analysis of archives, legal documents, interviews, and other textual evidence that demonstrated how the factors surrounding the Jan Kemp case evolved and how key administrators and faculty members reacted to pressure related to academic and athletic conflicts. An analysis of past reform efforts in college football identified presidential control, commercialization of athletics, and corruption of the student-athlete ideal through preferential treatment as the key issues universities must address in relation to their football programs. An analysis of the University of Georgia in relation to these issues showed that pressure to increase revenue from football led to a lack of presidential control over academic-athletic conflicts and allowed preferential treatment of athletes to persist at the expense of academic integrity.

La sélection génomique substitue à la connaissance de la généalogie celle des séquences d’ADN et connait un succès spectaculaire dans la sélection des bovins laitiers. En équin, le gain de précision pour les valeurs génétiques en CSO a été estimé faible entre la généalogie et la génomique, éventuellement à cause des particularités des populations d’apprentissage et de validation. L’objectif est de définir pour les races équines les conditions d’efficacité et de fonctionnement de la sélection génomique. La partie théorique de la thèse a consisté en une méta-analyse afin de comprendre le lien entre précision théorique et observée en fonction des paramètres des populations. L’étude a montré l’importance du nombre efficace de marqueurs Me. Ce paramètre spécifique de la population, de la structure génomique et de la parenté doit être évalué, au même titre que l’héritabilité en génétique classique. D’un point de vue pratique, la 1ère voie d’amélioration était de rechercher des gènes à effet majeur sur l’aptitude au concours de saut d’obstacles (CSO) ou au concours complet. Aucun gène majeur n’a été localisé malgré des détections significatives. Le 2nd levier pour améliorer l’estimation des valeurs génétiques en CSO était d’utiliser le Single-Step, méthode qui combine l’information génomique des étalons génotypés et la généalogie de l’ensemble des chevaux non génotypés utilisés pour l’indexation. L’évaluation pour le CSO a donc été revisitée. Malgré le re-calcul de l’héritabilité et l’application des points sur toute la période, le gain en précision reste faible. La sélection génomique a également été testée sur des chevaux d’endurance, mais comme pour le CSO les précisions obtenues pour le moment ne sont pas assez élevées pour justifier une utilisation de la sélection génomique. Récemment, un gène majeur agissant sur l’aptitude à trotter (DMRT3) a été identifié. Malgré l’effet très négatif d’un allèle sur la qualification et les performances précoces, le Trotteur français (TF) est polymorphe pour le gène à cause d’un effet positif de ce même allèle sur les performances tardives. La sélection classique et la sélection génomique ont été comparées en incluant ou non dans le modèle un marqueur lié à DMRT3, nous permettant d’identifier la meilleure combinaison de modèle et de méthode à utiliser pour estimer les valeurs génétiques du TF. Enfin, le paramètre Me a été estimé dans les populations de chevaux utilisées au cours de la thèse, et les résultats des évaluations génomiques ont été comparés en fonction de Me et des autres paramètres influant sur la précision de la sélection génomique. Deux nouveaux projets prévoyant de génotyper des chevaux de CSO d’une part et des TF d’autre part devraient permettre respectivement d’améliorer la précision de l’évaluation génomique en CSO et de confirmer l’intérêt de la prise en compte de DMRT3 dans l’évaluation génomique des TF. / Genomic selection uses genotypes information instead of pedigree information for the estimation of breeding values. In dairy cattle, the selection schemes were greatly improved with this method. In horses, a first attempt of genomic selection showed that the evaluation accuracy was not much improved when using genotypes information compared to classic evaluation, possibly because of the structure of the reference and validation populations. The objective of the thesis was to define the theoretical and practical conditions for the use of genomic selection in horses. The theoretical work of the thesis consisted in a meta-analysis to understand the relation between observed and theoretical accuracy depending on the parameters of the population. We proved the importance of the effective number of independent segments in the genome Me. This parameter is specific of the population and of the genomic structure and relationship structure. We recommend to estimate this parameter before genomic evaluation, just like heritability that is estimated before genetic evaluation. Regarding practical tasks of the thesis, the first solution to improve the breeding values estimation for jumping performances was to look for genes having a major effect on performances in jumping competitions and three-day’s events, but no major gene was evidence in spite of significant detections. The 2nd solution was to perform a single-step evaluation. This method combines information from genotyped stallions and from the pedigree of the whole population. Even if the heritability was re-estimated and points distributed to all horses to have a homogeneous criteria, the accuracy of genomic evaluation was not much improved. Genomic selection was also tested on horses running endurance races, but as for jumping the accuracy was not high enough. Recently, a major gene having a huge effect on the ability of horses to trot was evidenced (DMRT3). Even if one allele has a negative effect on qualification and early earnings, French Trotter (FT) is still heterozygote because of a positive effect of this allele on late performances. Genetic and genomic evaluations were compared with or without using in the model a SNP linked to DMRT3 as a fixed effect. This study allowed identifying the best combination of model and method to use for estimation of FT breeding values. Finally, the parameter Me was estimated in the populations of horses used in the thesis. The results of genomic evaluations were compared according to Me and the other parameters having an influence on the accuracy of genomic evaluations. Two new projects will genotype more jumping horses and FT, they should allow to improve the accuracy of genomic evaluation for jumping horses and to acknowledge the interest of using DMRT3 in the genomic evaluation of FT.

Abstract Iris The power of quantum computing lies in its ability to perform certain calculations and solve complex problems exponentially faster than classical computers. This potential has profound implications for a wide range of fields, including particle physics. This thesis lays a fundamental foundation for understanding quantum computing. Particular emphasis is placed on the intricate process of quantum gate decomposition, an elementary lynchpin that underpins the development of quantum algorithms and plays a crucial role in this research. In particular, this concerns the implementation of quantum algorithms designed to simulate the dynamic evolution of multi-particle quantum systems - so-called Hamiltonian simulations. The concept of quantum gate decomposition is introduced and linked to quantum circuit optimisation. The decomposition of quantum gates plays a crucial role in fault-tolerant quantum computing in the sense that an optimal implementation of a quantum gate is essential to efficiently perform a quantum simulation, especially for near-term quantum computers. Part of this thesis aims to propose a new explicit tensorial notation of quantum computing. Two notations are commonly used in the literature. The first is the Dirac notation and the other standard formalism is based on the so-called computational basis. The main disadvantage of the latter is the exponential growth of vector and matrix dimensions and the fact that it hides some relevant quantum properties of the operations by increasing the apparent number of independent variables. A third possible notation is introduced here, which describes qubit states as tensors and quantum gates as multilinear or quasi-multilinear maps. Some advantages for the detection of separable and entangled systems and for measurement techniques are also shown. Finally, this thesis demonstrates the advantage of quantum computing in the description of multi-particle quantum systems by proposing a quantum algorithm to simulate collective neutrino oscillations. Collective flavour oscillations of neutrinos due to forward neutrino-neutrino scattering provide an intriguing many-body system for time evolution simulations on a quantum computer. These phenomena are of particular interest in extreme astrophysical settings such as core-collapse supernovae, neutron star mergers and the early universe. A detailed description of the physical phenomena and environments in which collective flavor oscillations occur is first reported, and the derivation of the Hamiltonian governing the evolution of flavor oscillations is detailed. The aim is to reproduce this evolution using a quantum algorithm. To manage the computational complexity, we use the Trotter approximation of the time evolution operator, which mitigates the exponential growth of circuit complexity. The quantum algorithm was designed to work on a trapped-ion based testbed (the theory of which is presented in detail). After machine-aware optimisation, the quantum circuit implementing the algorithm was run on the real quantum machine 'Quantinuum', and the results are presented and discussed.

The Landau-Zener-Stückelberg-Majorana (LZSM) problem models diabatic transitions between energy levels in quantum two-level systems with an avoided level-crossing. The diabatic transition is a consequence of quantum tunneling in energy space when the system's Hamiltonian is perturbed with a fast-acting bias. The probability of transition between the energy states for a linear bias is known as the LZSM transition probability. The objective of this work is to investigate the LZSM problem through analytical and numerical lenses. The LZSM transition probability is derived in two ways. The first approach is based on Majorana's solution using contour integrals. The second derivation follows Landau's quasi-classical treatment. The derivations demonstrate methods for transitions in the presence of time-dependent perturbations. The ubiquity of the two-level system is discussed and an application on qubits concerning LSZM interferometry is presented, with the latter arising after considering periodic biases. Lastly, a simulation of the two-level system is conducted using Trotter-decomposed time-evolution operators, perturbation theory, and vectorization. The simulated transition probabilities for linear and periodic biases are obtained for varied parameters. The results show that the simulation achieves an accurate and efficient emulation of the LZSM problem.

Στην συγκεκριμένη εργασία ασχολούμαστε με την μελέτη θερμικά σύμπλεκτων (entangled) καταστάσεων πολλών κβαντικών bit (qubit) σε διάφορα μοντέλα Heisenberg με την μέθοδο Monte Carlo (MC). Αρχικά χρησιμοποιώντας τον μετασχηματισμό Suzuki-Trotter μετατρέπουμε την κβαντική μονοδιάστατη αλυσίδα των spin (μοντέλα Ising, Heisenberg με και χωρίς μαγνητικό πεδίο στις διευθύνσεις x,y,z) σε κλασικό δισδιάστατο πλέγμα. Εξετάζουμε την συμπεριφορά του συγκεκριμένου μετασχηματισμού για το αντισιδηρομαγνητικό Heisenberg ΧΧΧ μοντέλο, για το σιδηρομαγνητικό Heisenberg μοντέλο (ΧΧΧ και ΧΥΖ) χωρίς και με μαγνητικό πεδίο στις διευθύνσεις x,y,z για διάφορα μήκη της αλυσίδας, διαφορετικές διαστάσεις Trotter και διαφορετικό αριθμό Monte Carlo βημάτων (MCΒήματα). Μελετάμε την συμπεριφορά της θερμοχωρητικότητας, της ενέργειας, της μαγνητικής επιδεκτικότητας και της μαγνήτισης στις διευθύνσεις x,y,z. Επιβεβαιώνουμε την σωστή συμπεριφορά τους με βάση τα αναλυτικά αποτελέσματα. Τέλος γνωρίζοντας, ότι η κλασική συσχέτιση είναι το κάτω όριο της ποσότητας Localizable Entanglement, και ότι η ποσότητα Entanglement of Assistance είναι το πάνω όριο, εκτιμούμε για τα ίδια μοντέλα τη συμπεριφορά των ορίων και προσπαθούμε να εκτιμήσουμε το μήκος σύμπλεξης (Entanglement Length) για διάφορες θερμοκρασίες. / In the present Master Thesis we study the thermal entangled states of many qubits in a variety of Heisenberg models with the deployment of the Monte Carlo(MC) method. Initially we are using the Suzuki-Trotter decomposition in order to convert the one dimensional spin chain(models Ising, Heisenberg with and without magnetic field in the x,y,z axis) into a classical two dimensional lattice. We examine the behavior of the latter decomposition for the antiferromagnetic Heisenberg XXX model, the ferromagnetic Heisenberg model (XXX and XYZ) with or without magnetic field in the axis x,y,z for different chain lengths, Trotter dimensions and number of Monte Carlo Steps (MCSteps). We investigate the behavior of the following quantities: specific heat, energy, susceptibility and magnetization in the axis x,y,z. We confirm their proper behavior comparing to analytical and arithmetic results. Finally knowing that the maximum classical correlation function is the lower limit of the quantity Localizable Entanglement (LE) and that the quantity Entanglement of Assistance is the upper limit, we evaluate for the same models the behavior of the limits and we try to evaluate the Entanglement Length for a variety of temperatures.

Title: Quantum computing algorithms for quantum chemistry Author: Jakub Višňák Abstract: The topic of this study is the simulation of the quantum algorithm for the diagonalization of the matrix representation of the all-electron Dirac-Coulomb hamiltonian of the SbH molecule. Two different limited CI expansions were used to describe both the ground state (X 0+ ) and the first excited doublet (A 1) by simulating the Iterative Phase Estinamtion Algorith (IPEA). In the simulations numerically performed in this work, the "compact mapping" has been employed for the representation of the evolution operator exp(i Hˆ t); in the theoretical part of the work, the "direct mapping" is described as well. The influence of the metodics for choosing the initial eigenvector estimate is studied in both IPEA A and IPEA B variants. For those variants, the success probabilities pm are computed for different single-points on the SbH dissociation curves. The initial eigenvector estimates based on the "CISD(2)" method are found to be sufficient for both studied LCI-expansions up to internuclear distance R  6 a0. The pm dependence on the overlap between the eigenvector in question and its inital estimate - 2 0  is studied the for IPEA B method. The usability of the both variants of the IPEA in possible later calculations is...

<p>The recent groundbreaking demonstrations of quantum supremacy in noisy intermediate scale quantum (NISQ) computing era has triggered an intense activity in establishing finer boundaries between classical and quantum computing. In this dissertation, we use established techniques based on quantum Monte Carlo (QMC) to map quantum problems into probabilistic networks where the fundamental unit of computation, p-bit, is inherently probabilistic and can be tuned to fluctuate between ‘0’ and ‘1’ with desired probability. We can view this mapped network as a Boltzmann machine whose states each represent a Feynman path leading from an initial configuration of q-bits to a final configuration. Each such path, in general, has a complex amplitude, ψ which can be associated with a complex energy. The real part of this energy can be used to generate samples of Feynman paths in the usual way, while the imaginary part is accounted for by treating the samples as complex entities, unlike ordinary Boltzmann machines where samples are positive. This mapping of a quantum circuit onto a Boltzmann machine with complex energies should be particularly useful in view of the advent of special-purpose hardware accelerators known as Ising Machines which can obtain a very large number of samples per second through massively parallel operation. We also demonstrate this acceleration using a recently used quantum problem and speeding its QMC simulation by a factor of ∼ 1000× compared to a highly optimized CPU program. Although this speed-up has been demonstrated using a graph colored architecture in FPGA, we project another ∼ 100× improvement with an architecture that utilizes clockless analog circuits. We believe that this will contribute significantly to the growing efforts to push the boundaries of the simulability of quantum circuits with classical/probabilistic resources and comparing them with NISQ-era quantum computers. </p>